... the microphone Jack is on the front of the computer and not the monitor!

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erm, you didn't try to plug the headphone jack into the microphone socket did you? You may simply have to replace the speaker jack with the headphones jack unless your soundcard supports both inputs and have the software facility to mute speakers when the headphones are plugged in (as the Creative Soundblaster cards do!).

1 - Test the headphones on something you know works - establish that they work
2 - Ensure you are plugging the headphones into the Headphone socket on teh Dell - as Brassneck suggests - and NOT the Mic input.
3 - Sometime, there's a mute button/key combination on your PC. Check you haven't set that as muted
4 - Have you turned the volume up on the PC...? If you are using say iTues, ensure the volume slider is up
5 - Even if you have the volume slider up in your music player (iTunes/WinAmp, Media Player) - the system volume itself may be set too low.